3 Answers
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There isn't really a "gamification system" similar to how there isn't a "design system". There are different problems you can solve and different ways to solve them, and there are patterns and conventions that have worked well for others that you can learn from, sure. Your first link at gamification.org lists a bunch of them.

The most important thing you should realise about "gamification" is that it's a buzzword. Sort of like "Web 2.0". Gamification shouldn't be a goal. Instead, your goal should be delighting users, making things easier to use, or incentivising certain behaviour. Applying game mechanics to your design can be a great way to motivate users or provide incentives for them to do what you want them to do. But so can great visual design, writing, branding, etc.

In terms of finding different ways to approach applying game mechanics, gamification.org is a great resource. But don't just go there and stick achievements in your app. First identify what problem you want to solve and then find the relevant way to do that. Stephen Anderson's Mental Notes are a great tool you can use to help you do that.

If I can add to this, look at the system you're applying gameplay principles to, and find elements within there to create gameplay around. A good starting point is to identify the elements that create or are assigned values - we can quickly apply gameplay mechanics to things that produce a number or a comparison.

Another technique that Stephen Anderson refers to is picking an existing game (this can be anything from a card game to a Facebook game), listing the principles of gameplay behind it, and seeing if these principles can be applied to the system you're working with.

I haven't had a chance to try this approach but I think it's a brilliant way of identifying suitable gameplay mechanics for a system which doesn't naturally suggest gameplay.

Rahul is correct, gamification isn't a system to apply, it's really another set of design strategies, with all the artistry, pondering, and tradeoffs that that implies.

However a currently popular set of game mechanics (that are powerful, but often incorrectly used) are points, levels, badges, trophies, and virtual currencies.

A great way to start is to ask "what game are my users already playing?"

Usually there is some emotional reason people already engage is this type of experience. Once you find that, figure out a simple, clear, visually appealing way to display their progress in achieving that emotional goal (perhaps points). Then show progress toward milestones (maybe with levels) and celebrate their completion with rewards (badges, trophies, power, access, etc.)

This is just the tip of the iceberg, but should get you started in the right direction.

(You should also study successful games or gamified experiences that are as similar to what you're working on as possible.)